Unit 2079: Engineering Mathematics

1. Introduction to Engineering Mathematics

• Importance of mathematics in engineering disciplines
• Overview of basic mathematical concepts (sets, functions, algebraic expressions)
• Applications of mathematics in mechanical, electrical, civil, and computer engineering

2. Linear Algebra for Engineers

2.1 Matrices and Matrix Operations

• Definition and types of matrices
• Addition, subtraction, multiplication
• Transpose and inverse of matrices

2.2 Vectors and Vector Spaces

• Vectors, vector operations
• Basis, dimension, and linear independence

2.3 Systems of Linear Equations

• Solving linear systems using matrix methods
• Gaussian elimination

2.4 Determinants

• Properties and calculation
• Applications in system solvability

2.5 Eigenvalues and Eigenvectors

• Concepts and calculation
• Diagonalization
• Applications in stability analysis and vibrations

3. Calculus in Engineering

3.1 Differential Calculus

• Limits and continuity
• Derivatives and rules of differentiation
• Optimization problems in engineering

3.2 Integral Calculus

• Definite and indefinite integrals
• Techniques of integration
• Applications: area under curves, center of mass, work done

4. Differential Equations in Engineering

4.1 Ordinary Differential Equations (ODEs)

• First-order ODEs: separable, linear, exact equations
• Higher-order linear ODEs
• Applications: circuits, mechanical vibrations

4.2 Partial Differential Equations (PDEs)

• Introduction and classification
• Heat equation, wave equation, Laplace’s equation
• Engineering applications in heat transfer and fluid dynamics

5. Probability and Statistics for Engineers

5.1 Probability Theory

• Basic concepts: sample space, events, probability axioms
• Conditional probability and Bayes’ theorem

5.2 Probability Distributions

• Discrete and continuous distributions (Binomial, Normal, Poisson)

5.3 Statistical Analysis

• Descriptive statistics: mean, variance, standard deviation
• Inferential statistics: confidence intervals, hypothesis testing

5.4 Regression Analysis

• Simple linear regression
• Application in engineering data modeling

6. Numerical Methods in Engineering

6.1 Numerical Solutions of Equations

• Root finding methods: bisection, Newton-Raphson

6.2 Interpolation and Approximation

• Polynomial interpolation, splines

6.3 Numerical Differentiation and Integration

• Finite difference methods
• Trapezoidal and Simpson’s rules

6.4 Numerical Solutions of Differential Equations

• Euler’s method, Runge-Kutta methods

7. Fourier Analysis and Transforms

7.1 Fourier Series

• Periodic functions representation
• Convergence and properties

7.2 Fourier Transforms

• Definition and properties
• Applications in signal processing

7.3 Laplace Transforms

• Definition and inverse transform
• Solving differential equations
• Applications in control systems

8. Optimization Methods in Engineering

8.1 Linear Programming

• Formulation and graphical method
• Simplex method overview

8.2 Nonlinear Optimization

• Unconstrained and constrained problems
• Lagrange multipliers

8.3 Applications

• Engineering design optimization
• Resource allocation and decision making

9. Complex Variables in Engineering

9.1 Complex Numbers and Functions

• Algebra of complex numbers
• Analytic functions

9.2 Contour Integration

• Cauchy integral theorem and formula

9.3 Residue Theorem

• Calculation of residues
• Applications in evaluating integrals

9.4 Engineering Applications

• Fluid dynamics
• Electrical circuit analysis